On some possible generalizations of fractional Brownian motion

Fractional Brownian motion (fBm) can be generalized to multifractional Brow nian motion (mBm) if the Hurst exponent H is replaced by a deterministic fu nction H(t). The two possible generalizations of mBm based on the moving av erage representation and the harmonizable representation are first shown to be equivalent up to a multiplicative deterministic function of time by Coh en [S. Cohen, in: M. Dekking et al. (Eds.), Fractals: Theory and Applicatio ns in Engineering, Springer, Berlin, 1999, p. 3.] using the Fourier transfo rm method. Tn this Letter, we give an alternative verification of such an e quivalence based on the direct computation of the covariances of these two Gaussian processes. There also exists another equivalent representation of mBm, which is a variant version of the harmonizable representation. Finally , we consider a generalization based on the Riemann-Liouville fractional in tegral, and study the large time asymptotic properties of this version of m Bm. (C) 2000 Elsevier Science B.V. All rights reserved.